Preprint Article Version 1 This version is not peer-reviewed

Global Existence, Exponential Decay and Blow-Up of Solutions for a Class of Fractional Pseudo-Parabolic Equations with Logarithmic Nonlinearity

Version 1 : Received: 15 January 2018 / Approved: 22 January 2018 / Online: 22 January 2018 (04:22:52 CET)

How to cite: Liu, W.; Yu, J.; Li, G. Global Existence, Exponential Decay and Blow-Up of Solutions for a Class of Fractional Pseudo-Parabolic Equations with Logarithmic Nonlinearity. Preprints 2018, 2018010191 (doi: 10.20944/preprints201801.0191.v1). Liu, W.; Yu, J.; Li, G. Global Existence, Exponential Decay and Blow-Up of Solutions for a Class of Fractional Pseudo-Parabolic Equations with Logarithmic Nonlinearity. Preprints 2018, 2018010191 (doi: 10.20944/preprints201801.0191.v1).

Abstract

In this paper, we study the fractional pseudo-parabolic equations ut + (−△)s u + (−△)s ut = u log |u|. Firstly, we recall the relationship between the fractional Laplace operator (−△)s and the fractional Sobolev space Hs and discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence of global weak solution: for the low initial energy J(u0) < d, the solution is global in time with I(u0) > 0 or ∥u0X0(Ω) = 0 and blows up +∞ with I(u0) < 0; for the critical initial energy J(u0) = d, the solution is global in time with I(u0) ≥ 0 and blows up at +∞ with I(u0) < 0. The decay estimate of the energy functional for the global solution is also given.

Subject Areas

blow-up; fractional pseudo-parabolic equations; initial energy; logarithmic nonlinearity

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